TLDR; No, it cannot.
The answer to this question has a somewhat storied history.
Back in 1974 Wald considered a Gedanken experiment of what would happen if you would try to drop an object into a Kerr black hole that is at the extremal limit. He was able to prove that there are simply no geodesics that enter the black hole with sufficient J/E ratio to push the black hole past the extremal limit. This highlights the main physical obstacle to trying to "overspin" a Kerr black hole, to overspin you need to add mass with loads of angular momentum, but the centrifugal effect from the angular momentum makes it hard for the mass to enter the black hole.
Decades later, in 2009, Jacobson and Sotiriou considered the case closer to that of the OP. Starting from a nearly extremal Kerr black hole they showed that there is a small sliver of parameter space where geodesics entering the black hole have sufficient J/E ratio that they could seemingly push it past the extremal limit and overspin it. I say seemingly, because this analysis neglected the gravitational effects of the body falling into the black hole, the so-called gravitational self-force, which at the time nobody knew how to calculate.
A few years later in 2015 self-force calculations had made breakthrough allowing calculation of the GSF in Kerr spacetime. Using this new technology, Colleoni et al. showed in two papers that the effect of the GSF was just big enough to completely close this sliver of parameter space that would allow overspinning.
This answered the question for trying to overspin a Kerr black hole with a single particle/object. One might wonder (like the OP) of more general matter configurations could evade this result. Two years later Sorce and Wald came back and completely closed off this potential loop hole by showing no distribution of matter satisfying the null energy condition could ever overspin a black hole.*
Both results are marginal in the sense they exclude the possibility of overspinning, but do not exclude the possibility reaching extremality. Even the latter is expected to be impossible by the conjectured third law of black hole mechanics. However, technically, the also leave open the possibility that if one considers higher order corrections (e.g. the second order self-force) these would reopen the possibility of overspinning.
*Note both of Wald's proof's are more general and actually apply to general case of trying to push a Kerr-Newman black hole past extremality by adding charge and/or angular momentum. However, since the OP asked about Kerr I've limited this answer to just the Kerr case. For context it is fair to point out that the Jacobson and Sotiriou paper followed a longer discussion of overcharging nearly extremal Reissner-Nordström black holes initiated in a 1999 paper by Hubeny.
